Hidden truncation in multivariate Pareto (II) data: properties and inference

Abstract

In this follow up article of Arnold and Ghosh, we re-visit and study in details one of the two special cases of multivariate hidden truncation -(a) single variable truncation from above in a trivariate scenario where the component random variables follow an appropriate Pareto (type II) distribution. The case with single variable truncation was not discussed in details earlier in Arnold and Ghosh, in particular the estimation under both the classical and the Bayesian paradigm. It is well known that the Pareto distribution is a simple model for nonnegative data with a power law probability tail. Accumulating experience rapidly pointed out the fact that it is only in the upper tail of the income distributions that Pareto-like behavior can be expected. In practice, it has been observed that in income modeling and in other medical studies, quite often the observed data has been truncated with respect to several other unobserved co-variable(s) attaining certain critical levels. In this paper, we consider both the classical and the Bayesian inference (under the non-informative priors set up) of the resulting hidden truncated model. For illustrative purposes, a real data set on abalone is considered to show the flexibility of a single variable truncated Pareto (type II) distribution in a trivariate setting. Finally, the large sample behavior of the smallest order statistic in the case of a single co-variable truncation from above is provided in the appendix.